The short answer is that academies and free schools must act compatibly with human rights as set out in the Human Rights Act.
The Human Rights Act 1998 requires “public authorities” to “act compatibly” with “Convention rights”.
But again, I have seen academies, and even the Secretary of State reserve their position. See thus what the Secretary of State said in response to on judicial review:
“… the Secretary of State accepts for the purpose of the present proceedings that the Academy and the IAP [exclusions Independent Appeal Panel] are bodies which are amendable to Judicial Review and are public authorities for the purposes of the Human Rights Act 1998. This acceptance is for the present proceedings and should not be taken as an indication of the Secretary of State’s position in any other case.”
[I plan to say more here in due course about what that means in practice!]